One Knob, Three Sounds
A tone whose loudness wobbles. Speed the wobble up and it becomes three separate notes, without the signal changing in any way at all.
What it is
One tone, multiplied by a gain that wobbles. Drag across for how fast the wobble is, up and down for the pitch of the tone. It sweeps on its own until you touch it.
Three things happen as you go right, and only the last one is about the signal:
- on the left, a note getting louder and quieter, at a rate you could tap along to
- in the middle, a single rough, buzzing note
- on the right, three separate steady tones
The green line and the blue curve are where those changes are predicted to happen. Both come from measurements of the ear that have nothing to do with this signal.
How it works
Multiplying a cosine at f_c by (1 + m·cos 2πf_m t) gives exactly three sinusoids: the carrier, and one sideband either side at f_c ± f_m. Not approximately three. Exactly three.
Here’s the signal rendered and measured with Goertzel filters, at a 1000 Hz carrier and depth 0.8:
| modulation rate | lower sideband | carrier | upper sideband | anything else |
|---|---|---|---|---|
| 1 Hz | 0.4000 | 1.0000 | 0.4000 | 3 × 10⁻¹³ |
| 12 Hz | 0.4000 | 1.0000 | 0.4000 | 1 × 10⁻¹³ |
| 40 Hz | 0.4000 | 1.0000 | 0.4000 | 1 × 10⁻¹³ |
| 160 Hz | 0.4000 | 1.0000 | 0.4000 | 1 × 10⁻¹³ |
| 400 Hz | 0.4000 | 1.0000 | 0.4000 | 1 × 10⁻¹³ |
Identical to four decimal places the whole way along, with nothing anywhere else in the spectrum above 10⁻¹³. The envelope shape is identical too, and 24.2% of the power sits in the sidebands at every single rate.
Whatever is changing as you drag, it isn’t the signal.
What surprised me
I expected to predict one boundary. There are two, they come from different faculties, and only one of them moves.
The right-hand boundary is spectral. The sidebands sit rate either side of the carrier
so the pair spans twice the rate, and they become separately audible once that span
exceeds the carrier’s critical bandwidth. That number comes from masking experiments, so
I can look it up rather than fit it. Critical bands widen with pitch, so the boundary is
a curve:
| carrier | critical bandwidth | sidebands separate above | width of the rough band |
|---|---|---|---|
| 100 Hz | 101 Hz | 50 Hz | 30 Hz |
| 500 Hz | 117 Hz | 59 Hz | 39 Hz |
| 1000 Hz | 162 Hz | 81 Hz | 61 Hz |
| 4000 Hz | 685 Hz | 343 Hz | 323 Hz |
| 8000 Hz | 1706 Hz | 853 Hz | 833 Hz |
The left-hand boundary is temporal and sits at about 20 wobbles a second whatever the pitch, because it’s a limit on resolving events in time, not frequencies.
That gives you the bit I didn’t see coming. The rough middle is the gap between a fixed limit and a rising one, so it’s twenty-seven times wider at 8 kHz than at 100 Hz. The same 40 Hz modulation is three distinguishable tones on a low carrier and a single buzzing note on a high one. Drag straight up the page at a fixed rate and the sound changes category while the modulation doesn’t change at all.
Two boundaries, one signal, one knob, and neither boundary is in the sound. This is the cleanest version I’ve built of something this series keeps running into. The measurement you make of a signal and the thing a listener does with it are different objects, and the interesting facts are usually about the second one.
What I would do next
Sweep the depth down towards zero and find where each boundary stops existing. The sidebands have to become audible before they can become separable, so there should be a depth below which the rough region has no lower edge at all.